where \(| \cdot |\)  is a determinant, and \(\mathcal {F}(\theta )\)  is the Fisher Information Matrix, defined as

F()ij = -Eq(x;)[2ijq(x;)]  .

The Fisher Information Matrix is also the local metric on the statistical manifold with coordinates \(\theta \)  [1]}, it quantifies how “distinguishable” \(\theta \)  is from \(\theta +d\theta \) . Note that (REF ) can be viewed as the entropy of a Gaussian approximation to \({\rm q}(\hat{\theta }|\mathbf {x})\)  with precision matrix \(\mathcal {F}(\theta )\) ; this Gaussian approximation is most accurate when \({\rm q}(\mathbf {x};\theta )\)  itself is narrow and approximately Gaussian [2]}.
